English Translation
If you want to do a good job, don’t rush through it.
I’m curious about the prepositional phrase. It seems to mean something like “in regard to”. Is “auf” used this way with other concepts?
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You are right. “Auf die Schnelle” is really a strange expression.
“Schnelle” as a noun of “schnell” is not used in any other context.
“Auf” is used here similar to “auf diese Weise / auf diese Art” = “in this way”.
There are some similar constructions in other colloquialisms, e.g.
“Auf Deuvel komm raus” (lit. on that the devil might appear) =“come hell or high water”
“Auf Biegen und Brechen” (lit. on that it may bend or break") =“it’s make or break”)
“Auf Gedeih und Verderb” (lit on that it may prosper or perish) =“root, hog or die”.
Please note that my translations are just proposals, that should get “the spirit” of the expressions across.
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not to forget “Auf Wiedersehen” 
You could also say “Auf unsere Freundschaft” (here’s to our friendship) when toasting. I think this is actually “wir trinken auf unsere Freundschaft”.
There aren’t many that come to mind…
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Yes, of course. “Auf Wiedersehen.” How could it forget?
The simplest example is the best. 
There aren’t many that come to mind…
In the meantime I’ve come up with a few more examples with auf:
Ich wollte mal auf einen Kaffee vorbeischauen. - I wanted to drop by for a coffee.
Auf ein Wort! - Let’s talk! / We need to talk.
Auf ein Neues. - Let’s try/do it again.
Auf dass ihr immer glücklich seid. - (obsolete syn. of damit - so that) May you always be happy.
Auf keinen Fall! - No way! / Under no circumstances. / That is out of the question.
Auf jeden Fall! - Absolutely!
Auf freiem Fuß(e) - to be on the loose
Auf einmal - suddenly
Ach, auf einmal fällt’s dir wieder ein. - Oh, now you remember! (I know you knew it all along, you just pretended not to know it)
Auf gut Glück! - Google’s search button “I’m feeling lucky”
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